A Classification of Spherical Curves Based on Gauss Diagrams

We consider generic smooth closed curves on the sphere
$S^{2}$. These curves (oriented or not) are classified relatively to
the group $\mbox{Diff}(S^{2})$ or its subgroup $\mbox{Diff}^{+}(S^{2})$),
with the Gauss diagrams as main tool. V. I. Arnold determined the
numbers of orbits of curves with $n$ double points when $n<6$. This
paper explains how a preliminary classification of the Gauss
diagrams of order 5, 6 and 7 allows to draw up the list of the
realizable chord diagrams of these orders. For each such diagram
$\Gamma$ and for each Arnold symmetry type $T$, we determine the
number of orbits of spherical curves of type $T$ realizing $\Gamma$.
As a consequence, we obtain the total numbers of curves (oriented or
not) with 6 or 7 double points on the sphere (oriented or not) and
also the number of curves with special properties (e.g. having no
simple loop).

In this paper, a spherical curve will be a generic smooth
closed curve on $S^{2}$, more explicitly the image of the unit circle
under an immersion $f:S^{1}\rightarrow S^{2}$, the multiple points of
which are double points $f(a)=f(b)$ ($a\neq b$) with distinct
tangent directions.

We consider not only spherical curves (often denoted by $C$) but
also oriented spherical curves (denoted by $C^{+}$ and $C^{-}$). Also
the sphere may be oriented; this allows to define
orientation-preserving $C^{\infty}$-diffeomorphisms of $S^{2}$; the
group of such diffeomorphisms will be denoted by
$\mbox{Diff}^{+}(S^{2})$. When $G$ is a group acting on a set $X$, one
says that two objects $U,V$ are $G$-equivalent if there is
an element of $G$ mapping $U$ onto $V$. In particular, it makes
sense to consider $\mbox{Diff}(S^{2})$-equivalent or
$\mbox{Diff}^{+}(S^{2})$-equivalent curves or oriented curves.

We take an interest in the classifications of spherical curves. To
begin with, we consider curves without double points. It is known
that two oriented spherical curves without double point are
$\mbox{Diff}^{+}(S^{2})$-equivalent and, a fortiori, that two spherical
curves without double point are $\mbox{Diff}^{+}(S^{2})$-equivalent. An
analogous property is true for oriented curves with just one double
point, but not for oriented curves with more than one double point:
the number of $\mbox{Diff}^{+}(S^{2})$-orbits is then larger than 1.
Results of V.I. ([Arnold1994], p. 27) over the numbers of orbits of
spherical curves with at most five double points are summarized in
Table 1.

Table 1: Numbers of orbits of spherical curves found by Arnold

Oriented object(s)

n = 0

1

2

3

4

5

$S^{2}$, curves

1

1

3

9

37

182

$S^{2}$

1

1

2

6

21

99

curves

1

1

2

6

21

97

none

1

1

2

6

19

76

The last row contains the numbers of orbits of unoriented spherical
curves with $n$ double points for $n$ in $\{$0, 1,…, 5$\}$,
relatively to the $\mbox{Diff}(S^{2})$-equivalence; the other rows
contain the numbers of orbits when $S^{2}$ or $S^{1}$ are oriented.

The method used by [Arnold1996] for computing the numbers of
orbits of spherical curves becomes very long when the number $n$ of
double points is larger than 5; this prompted us to look for another
way to compute them. The method presented in this paper is
essentially based on a structured classification of combinatorial
objects, the Gauss diagrams, which summarize the intrinsic geometry
of the curves (one forgets that the curves are contained in a
sphere). With this method, we first verified the correctness of the
values in Table 1, and afterwards we found the values for
$n=6$ and $n=7$, given in Table 2.

With Arnold we shall say that a curve is irreducible if it
cannot be disconnected by removing one double point. As they are
building blocks for all other curves, it is important to know them
concretely. [Arnold1996] described 9 irreducible curves with 7
double points, represented in Fig. 1.

Figure 1: Arnold’s irreducible curves with 7 double points

To get a complete list of representatives of all irreducible curves
with 7 double points, one must add four elements, represented in
Fig. 2.

Figure 2: Supplementary irreducible curves with 7 double points

No part of an irreducible curve is a simple loop (closed curve
which is $\mbox{Diff}(S^{2})$-equivalent with the right part of the
symbol infinity). If we are looking for representatives of all
spherical curves with 7 double points and without any simple loop,
we must add to the curves of Figs. 1 and 2 the curves of Fig. 3.

Let $\gamma:S^{1}\rightarrow S^{2}$ be an immersion whose image is a
curve $C$ with $n$ double points and let $D$ be the closed disk
bounded by $S^{1}$; for every pair ${a,b}$ of points of $S^{1}$ which
determine a double point of $C$, one may draw a line segment joining
$a$ and $b$; the figure consisting of the disk $D$ and these $n$
line segments is the Gauss diagram $\Gamma(\gamma)$. The $n$
segments are the chords and their 2$n$ endpoints are the nodes of $\Gamma(\gamma)$.

As we are interested in the curves $\gamma(S^{1})$ and not of the
immersions $\gamma$, it is not a restriction to suppose that the
nodes of every Gauss diagram are regularly placed on $S^{1}$ (they are
the vertices of a regular $2n$-gon).

As a matter of fact, the Gauss diagram of $\gamma$ is a graphical
representation of a combinatorial object related with the curve
$\gamma(S^{1})$ which is defined below when $n$ is an integer larger
than 1 (there are minor changes when $n$ = 0 or 1).

Definition.

A chord diagram of order $n$ is a set $\mathcal{A}$ of 2$n$
elements, the $nodes$, provided with two families $\mathcal{E}$ and
$\mathcal{C}$:

(a)

The elements of
$\mathcal{E}$
, called
$\it edges$
, are
pairs of nodes such that
$(\mathcal{A},\mathcal{E})$
is a circuit
graph (if the chord diagram comes from
$\Gamma(\gamma)$
, the edges
come from endpoints of chords which are neighbours on
$S^{1}$
);

(b)

The elements of
$\mathcal{C}$
, called chords, are
disjoint pairs of nodes whose union is
$\mathcal{A}$
(if the chord
diagram comes from
$\Gamma(\gamma)$
, the elements of
$\mathcal{C}$
come from the chords of
$\Gamma(\gamma)$
).

As $(\mathcal{A},\mathcal{E})$ is a connected graph, two nodes have
a distance: the smallest number of edges needed to join the nodes.
In a chord diagram, the $\it step$ of a chord $ab$ is the distance
between $a$ and $b$. The following property of $\Gamma(\gamma)$ was
observed by [Gauss1900]: along the arcs of $S^{1}$ limited by
endpoints of every chord, the number of nodes is always even; this
suggests a

Definition.

A Gauss diagram of order n is a chord diagram of order $n$
satisfying the parity condition: the step of every chord is
odd.

For example, the first figure below represents a Gauss diagram, but
the second one does not:

Isomorphic Gauss diagrams will be identified; so there is only one
Gauss diagram of order 0, one of order 1 and one of order 2, but three
of order 3.

Remark.

A Gauss diagram of order $n$ is not necessarily related with a
generic spherical curve having $n$ double points; this Gauss
realizability problem was solved on different ways in papers of [Francis1969], [Marx1969], [Lovasz and Marx1976], [Rosenstiehl and Read1977], [Dowker and Thistlethwaite1983], [Chaves and Weber1994].

For example, the following Gauss diagrams are not realizable:

Definition.

The distinct chords $ab$ and $cd$ of a Gauss diagram $\Gamma$ are
interlaced if $a,c,b,d$ are in cyclic order along the circuit
graph of $\Gamma$; with Rosenstiehl and Read, we define the interlacing graph of $\Gamma$ as follows: its nodes are the chords
of $\Gamma$ and its edges are the pairs of interlaced chords.

New Look of Diagrams: if one represents chords of a Gauss diagram by
chords of a circle, the chords the step of which is small w.r.t. the
order of a Gauss diagram are not very visible; for this reason and
for aesthetic care, we replace the line segments by circular arcs in
the drawing of chords. For example, the diagram showed below on the
left has the new look drawn on the right:

We are now in a position to define the families of Gauss diagrams;
we begin with the notion of fatherhood for diagrams.

Definition.

If the Gauss diagram $\Gamma=(\mathcal{A},\mathcal{E},\mathcal{C})$ has at least one chord the
step of which is 1, the father-diagram of $\Gamma$ is the
diagram $\Gamma^{\prime}=(\mathcal{A^{\prime}},\mathcal{E^{\prime}},\mathcal{C^{\prime}})$
obtained as follows:

(a)

$\mathcal{C}^{\prime}$
is the set of the chords of
$\Gamma$
whose
step is larger than 1;

(b)

$(\mathcal{A^{\prime}},\mathcal{E^{\prime}})$
is the circuit graph
obtained by deleting the nodes which are endpoints of chords of
$\Gamma$
with step 1, and respecting the cyclic order of the
preserved nodes (it is the empty graph if
$\mathcal{C}^{\prime}$
is empty).

For example, the father-diagram of the Gauss diagram of order 6
represented below by the first drawing is the Gauss diagram of order
4 represented by the second one, which is equivalent to the third
one:

When $\Gamma^{\prime}$ has at least one chord with step 1, it has a
father-diagram. The transfer from a diagram $\Gamma$ to its
father-diagram may be iterated until one gets a diagram without
chord with step 1; the latter is called leader-diagram of
$\Gamma$.

For example, the leader-diagram of the Gauss diagram showed below on
the left is the Gauss diagram represented on the right:

Definition.

The family of a Gauss diagram $\Gamma$ is the (infinite) set
of Gauss diagrams which have the same leader-diagram as $\Gamma$.
The natural bijection between the set of the just defined families
and the set of the leader-diagrams allows us to use the same drawing
to denote a family and its leader-diagram.

The following properties, easy to prove, show that the concept
family of diagrams is natural, especially for a classification of
spherical curves.

1.

The relation of fatherhood turns each family into a
rooted tree. Three such trees are partially represented in Fig. 4.

2.

All Gauss diagrams of a family have interlacing
graphs with the same number of edges. More precisely, the
interlacing graph of a diagram
$\Gamma$
is isomorphic to the union
of a totally disconnected graph and the interlacing graph of the
leader-diagram of the family of
$\Gamma$
.

3.

If a Gauss diagram is realizable, then all the Gauss
diagrams of its family are also realizable; hence one can say, in
this case, that the family is realizable.

The way of constructing the Gauss diagrams of order $n>$ 4 combines
two methods: the first one gives the list NL(n) of
the Gauss diagrams which are not leader-diagrams and is based on the
catalogue of all Gauss diagrams whose orders are smaller than $n$;
the second one gives the catalogue $L(n)$ of the leader-diagrams,
and uses a part of the list NL(n). We describe the
first method immediately and delay the second one to
Sect. 3.

If $\Gamma$ belongs to NL(n), it has a
father-diagram which belongs to the catalogue of all Gauss diagrams
whose orders are $<n$. Hence one will obtain the elements of
NL(n) by answering, for every Gauss diagram
$\Gamma^{\prime}$ of order $<n$, the question:

what are the Gauss diagrams of order $n$ that have $\Gamma^{\prime}$
as father-diagram?

Let us consider the case where $n=$ 5 and $\Gamma^{\prime}$ is the Gauss
diagram of order 3 represented hereunder; we must augment $\Gamma^{\prime}$
by means of two chords with step 1.

Let us try to put a first chord between $b$ and $c$, and a second
one between $d$ and $e$. The result will be a diagram of order 5
whose father-diagram is the unique diagram of order 2 and not
$\Gamma^{\prime}$.

To avoid this contradiction, it is necessary that the chords with
step 1 in $\Gamma^{\prime}$ do not remain chords with step 1 in the new
diagram; with other words:

At least one “new” chord with step 1 must be put in each edge
of $\Gamma^{\prime}$ joining the endpoints of an “old” chord with step 1.

Obeying this rule and using the symmetries of $\Gamma^{\prime}$, one gets
the complete list NL(n) without a double
occurrence.

In the rest of this text, we use the abbreviation $k$-chord
for chord with step equal to $k$. By adding 1-chords to Gauss
diagrams of orders $<$ 5, as explained above, one gets the list
NL(5) that appears in Table 3.

Table 3: The 13 Gauss diagrams of order 5 which are not
leader-diagrams

The partial representation of the rooted trees drawn in Fig. 4 may
now be extended to order 5; Fig. 5 shows the result for the second
and the third families.

Figure 5: Parts of the rooted trees starting with the
leader-diagrams of order 3 and 4

We now explain a method that may be applied to construct all
leader-diagrams of a given order $n$ $(n>3)$; for clarity, we expose
it for $n=5$.

Among the elements of NL(5), there are three Gauss
diagrams without 3-chord; we now label their nodes by the integers
modulo 10, in such a way that the neighbours of the node $k$ are the
nodes $k-1$ and $k+1$.

Let $\Gamma$ be any diagram above. As 3 is invertible in the ring
$\mathbb{Z}/10\mathbb{Z}$, the multiplication by 3 defines a
permutation $m_{3}$ of the nodes of $\Gamma$; we now replace the
chords of $\Gamma$ by their images under $m_{3}$; each 1-chord becomes
a 3-chord and each 5-chord remains a 5-chord. So this construction
provides 3 elements of $L(5)$:

All Gauss diagrams with at least one 3-chord and without 1-chord are
obtained by this procedure. In order to complete the list $L(5)$, it
suffices to add the Gauss diagrams of order 5 which have neither
1-chord nor 3-chord; but this is easy: such a diagram is unique:

We proceed as for the Gauss diagrams of order 5, constructing first
the elements of NL(6); they are shown in the second
column of Table 4.

Table 4: The 44 Gauss diagrams of order 6 which are not
leader-diagrams

Among the 44 elements of NL(6), seven diagrams have no
5-chords; they bear the numbers 1, 2, 7, 8, 9, 13 and 21. If we
label their nodes with the elements of the ring
$\mathbb{Z}/12\mathbb{Z}$, and proceed as we did for the
leader-diagrams of order 5 (now using the multiplication $m_{5}$
instead of $m_{3}$), we obtain 7 elements of $L(6)$, namely

The elements of $L(6)$ which are not obtained in this way are the
Gauss diagrams whose the step of every chord is 3; they are showed
below (it is an easy exercise to see that there is no other).

We have applied the procedure described at the end of Sect. 2 for
drawing up the list of the elements of NL(7); as it is a
set of cardinality 217, we shall not annoy the reader with its
description. However, for the reader who would like to check this
cardinality, we decompose it as follows: the families

respectively include 27, 48, 34, 11, 21, 21, 7 elements of
$NL(7)$ and the union of the families generated by leader-diagrams
of order 6 includes 48 elements of $NL(7)$.

The set NL(7) contains 37 elements which have no 5-chord;
these may be transformed (using the function $m_{3}$ in the ring
$\mathbb{Z}/14\mathbb{Z}$) into diagrams without 1-chord, but having
at least one 3-chord. The result gives a first part of $L(7)$:

The other elements of $L(7)$ have neither 3-chord, nor 1-chord; if
they have at least one 5-chord, they may be obtained (using the
multiplication $m_{5}$) from the elements of NL(7) having
neither 5-chord, nor 3-chord. The result gives a second part of the
set $L(7)$:

The last element of $L(7)$ is the diagram of order 7 which has only
7-chords.

The parity condition imposed in our definition of a Gauss diagram
$\Gamma$ of order $n$ is not sufficient to insure that $\Gamma$ is
realizable, i.e. is the Gauss diagram of a spherical curve. A second
necessary condition of realizability is the

Biparity condition: If the chords $aa$’ and
$bb$’ of $\Gamma$ are not interlaced, then the number of
chords which are interlaced with $aa$’ and with $bb$’ is
even.

Here is an example of a diagram that does not verify the biparity condition:

One notices that, if a Gauss diagram $\Gamma$ satisfies the biparity
condition, then all the diagrams in the family of $\Gamma$ also
satisfy it. In this case, one can say that the family of $\Gamma$
satisfies the biparity condition.

Among the families the leader of which has an order smaller than 5
(drawn hereafter), the biparity condition is always verified.

Among the families whose leader has order 5, the biparity condition
is verified for half of the case, namely for the families

Among the families whose leader has order 6, the biparity condition
is verified for 5 cases out of 9; here are the leaders of the 5
families

Among the families whose leader has order 7, the biparity condition
is satisfied only in 14 instances out of 43; they are represented
hereafter:

It remains to answer the questions, for every diagram among the 24
above: is it a realizable Gauss diagram? In the affirmative, what
are the realizations?

As it happens, most of the 24 leader-diagrams described in the
preceding section are realizable. The exceptions are the diagram
$\Gamma_{6}$ of order 6 whose every chord has step 5 and the diagram
of order 7 numbered as 35.

We prove by contradiction that $\Gamma_{6}$ is not realizable: suppose
to the contrary that it is; then a vertex split at $a$ (see
definition and property in [Lovasz and Marx1976]) would
transform it into a realizable diagram $\Gamma_{5}$:

The contradiction is that $\Gamma_{5}$ does not verify the
biparity condition, so it cannot be realizable.

An analogous reasoning proves that also the diagram 35 is not
realizable.

Families generated by a realizable leader-diagram of order
$<$8.

In Table 5, we associate a capital letter with each of the
22 families generated by a realizable leader-diagram of order $<$8.

Table 5: Names of the families including a realizable diagram of order $<8$

One says that a Gauss diagram $\Gamma$ is decomposable if
there exists a partition $\left\{A,B\right\}$ of the set of chords
of $\Gamma$ such that no element of $A$ is interlaced with an
element of $B$; this determines a decomposition of $\Gamma$ into two
smaller diagrams; for example, the leader-diagram of family
V has a decompostion consisting in the leader-diagrams of
families B and C.

Let us say that a Gauss diagram has essentially one
realization if it is realizable and all its realizations belong to
the same $\mbox{Diff}(S^{2})$-orbit. Using the informations given by
any of the 22 realizable leader-diagrams of order not greater than
7, one comes to the following conclusions:

(a)

every leader-diagram of order $<$6 has essentially one realization,

(b)

one leader-diagram of order 6 has two orbits of
realizations, the other three just one,

(c)

all but two leader-diagrams of order 7 have essentially
one realization, one exception with two realizations and the other
exception with three.

Pictures of curves with $n$ double points ($n<8$) without
any simple loop (realizations of leader-diagrams of order $<$8).

In Figs. 6 and 7, we exhibit drawings of realizations in increasing values
of the order $n$ and, in case of equality of order, in decreasing
values of the number of edges $int$ of its interlacing graph.
Sometimes, we show two plane representations which are
$\mbox{Diff}(S^{2})$-equivalent; but in most cases, the chosen drawing
is a curve with a maximum number of symmetries.

Figure 6: Spherical realizations of the leader-diagrams of
order $<$7

Figure 7: Spherical realizations of the leader-diagrams of
order 7

Remark.

(1)

If
$n\in\{2,3,\ldots,7\}$
, the leader-diagrams of order
$n$
which
are not decomposable are exactly the prime diagrams of order
$n$
in
the paper of [[Chmutov et al.2006]].

(2)

The groups of diffeomeophisms preserving the realizations
of the leader-diagrams B, C, D and K act
transitively on their double points; the description of all
spherical curves having this property is given in our paper ([Valette2016]).

The spherical realizations of a Gauss diagram $\Gamma$ which has a
father-diagram could be done like those of the leader-diagrams, but
this method is lengthy. We prefer to make use of the realizations of
the father-diagram of $\Gamma$, which are supposed already known.
The idea is simple: in each $\mbox{Diff}(S^{2})$-orbit of realizations
of the father-diagram of $\Gamma$ we choose a model $M$, called
mother-curve in the sequel, and we proceed as follows: for each edge
of the father-diagram where some 1-chords were added to get
$\Gamma$, we mark, by means of a short stroke, the corresponding
open arc of $M$ which must be modify to give birth to a loop of the
realization of $\Gamma$ in progress; an analysis of the
combinatorial symmetries of $M$ enables to eliminate duplications of
the resulting curve. We give additional explanations on this
analysis by treating an example.

We first introduce some terminology: if $M$ is a spherical curve
with at least one double point, we define an $M$-domain $D$ as
a connected component of the complement of $M$ in $S^{2}$ and a side of $D$ as the closure of an open arc of the boundary of $D$
without double point and maximal for this property.

Imagine that we are looking for the spherical realizations of the
diagram $\Gamma$ showed below on the left. The father-diagram of
$\Gamma$, showed on the right,

may be realized by several $\mbox{Diff}(S^{2})$-equivalent
curves, for instance

(we denote a point of $S^{1}$ and its image through a
parametrization by the same letter). If $M$ is one of these models,
the stabilizer $H$ of $M$ in the group $\mbox{Diff}(S^{2})$ acts
transitively on the set of sides $ac,ad,bc$ and $bd$, so that we
may choose $ad$ as arc where a loop will be added. Moreover, the
model below, on the left, shows that there exists in $H$ a
diffeomorphism permuting the $M$-domains $adc$ and $dab$.

We may put the loop in $dab$. Therefore there is
essentially one spherical realization of the Gauss diagram $\Gamma$.
It looks like the curve sketched on the right.

The method explained above allows to determine, for any Gauss
diagram $\Gamma$, a complete set of spherical curves realizing
$\Gamma$ (such a set $S$ is complete if any spherical curve
with Gauss diagram $\Gamma$ is Diff$(S^{2})$-equivalent with
exactly one element of $S$). Table 6 gives, for every family whose
leader-diagram has order $<$7 and for every $n\leq 7$, the number of
curves with $n$ double points, up to
Diff$(S^{2})$-equivalence, which realize a Gauss diagram
belonging to the family.

Table 6: Number of curves with n double points, up
to Diff$(S^2)$ equivalence, which realize a Gauss diagram belonging to a
given family.

The Appendix concerning Table 6 shed light on the way to obtain the
given values: beside every non leader-diagram $\Gamma$ appears the
number of orbits of curves realizing $\Gamma$.

As a consequence, we obtain a classification of the spherical curves
based on the number of double points and the absence of simple loops (Table 7).

[Arnold1994]; [Arnold1996] distinguishes between five types of
symmetry for plane curves; this classification is used by [Gusein-Zade and Duzhin1998] for determining the numbers of
curves (oriented or not) with $n$ double points ($n\leq 10$) in the
plane (oriented or not).

We shall give an analogous classification of spherical curves, using
following notations:

(a)

If $C$ is a curve, we denote by $C^{+}$ and $C^{-}$ the
oriented curves along $C$;

(b)

We denote by $[C]$ the orbit of $C$ under
Diff${}^{+}(S^{2})$; we define analogously $[C^{+}]$ and $[C^{-}]$;

(c)

We denote by $-C$ the image of $C$ under the antipody of
$S^{2}$, that is the restriction to $S^{2}$ of the operator
$\mathbf{x}\rightarrow\mathbf{-x}$ in $\mathbb{R}^{3}$; we define
analogously the oriented curves $-C^{+}$ and $-C^{-}$. The orbit
$[-C^{+}]$ of $-C^{+}$ under Diff${}^{+}(S^{2})$ is also the set of the
images of $C^{+}$ under the elements of
Diff$(S^{2})\setminus$Diff${}^{+}(S^{2})$; hence the
equality $[f(C^{+})]=[-C^{+}]$ is valid for every oriented curve $C^{+}$
and every reflection $f$.

It is fairly obvious that the orbit of $C$ (resp. $C^{+}$, resp.
$C^{-}$) under Diff$(S^{2})$ is $[C]\cup[-C]$ (resp.
$[C^{+}]\cup[-C^{+}]$, resp. $[C^{-}]\cup[-C^{-}]$).

We now define the Arnold type of a spherical curve $C$ by looking at
the possible coincidences of the orbits $[C^{+}]$, $[C^{-}]$, $[-C^{+}]$,
$[-C^{-}]$. Because each equality of two among these orbits implies
the equality of the other two, the number of cases reduces to 5.

(1)

If the orbits $[C^{+}]$, $[C^{-}]$, $[-C^{+}]$, $[-C^{-}]$ are
different from each other, then we say that the curve $C$ is asymmetric or of type Asy; this implies that the oriented
curves $C^{+}$ and $C^{-}$ are not equivalent on the unoriented sphere
and that the curves $C$ and $-C$ are not equivalent on the oriented
sphere; in other words, the contribution of such a curve

to the number of orbits of oriented curves on the unoriented sphere
is 2,

to the number of orbits of unoriented curves on the oriented sphere
is 2,

to the number of orbits of oriented curves on the oriented sphere is
4.

A 4-tuple summarizes these properties: $(1,2,2,4)$, later used as a column of a matrix.

Three examples of curves of type Asy are given hereafter.

Remark that the last one is invariant under a half-turn and
nevertheless is of type Asy.

(2)

If the orbits $[C^{+}]$, $[C^{-}]$, $[-C^{+}]$ and $[-C^{-}]$
coincide, then we say that $C$ is supersymmetric or of type Sup; this implies that the oriented curves $C^{+}$ and $C^{-}$ are
equivalent on the unoriented sphere and that the curves $C$ and $-C$
are equivalent on the oriented sphere; in other words, the
contribution of $C$

to the number of orbits of oriented curves on the unoriented sphere is 1,

to the number of orbits of unoriented curves on the oriented sphere is 1,

to the number of orbits of oriented curves on the oriented sphere is 1.

The 4-tuple related to the type Sup is thus $(1,1,1,1)$.

Four examples of curves of type Sup are given herafter.

One easily sees that $[-C^{+}]=[C^{-}]$ for the last two curves
above; in order to be convinced that the equalities $[-C^{+}]=[C^{+}]$ and $[C^{-}]=[C^{+}]$ are also true for these curves, it is
convenient to look at the following models:

(3)

If $[-C^{+}]=[C^{-}]\neq[-C^{-}]=[C^{+}]$, then we say that
$C$ is symmetric(1) or of type Sy1; this implies that
the oriented curves $C^{+}$ and $C^{-}$ are equivalent on the unoriented
sphere and that the curves $C$ and $-C$ are equivalent on the
oriented sphere; with other words, the contribution of such a curve

to the number of orbits of oriented curves on the unoriented sphere is 1,

to the number of orbits of unoriented curves on the oriented sphere is 1,

to the number of orbits of oriented curves on the oriented sphere is 2.

The 4-tuple related to the type Sy1 is thus $(1,1,1,2)$.

For instance, the curves hereunder are plane representations of
spherical curves of type Sy1.

(4)

If $[C^{-}]=[C^{+}]\neq[-C^{-}]=[-C^{+}]$, then we say that
$C$ is symmetric(2) or of type Sy2; this implies that
the oriented curves $C^{+}$ and $C^{-}$ are equivalent on the unoriented
sphere and that the curves $C$ and $-C$ are not equivalent on the
oriented sphere; with other words, the contribution of such a curve

to the number of orbits of oriented curves on the unoriented sphere is 1,

to the number of orbits of unoriented curves on the oriented sphere is 2,

to the number of orbits of oriented curves on the oriented sphere is 2.

The 4-tuple related to the type Sy2 is thus $(1,1,2,2)$.

The curves hereafter are models of spherical curves of type Sy2.

(5)

If $[-C^{+}]=[C^{+}]\neq[-C^{-}]=[C^{-}]$, then we say that $C$ is
symmetric(3) or of type Sy3; this implies that the
oriented curves $C^{+}$ and $C^{-}$ are not equivalent on the unoriented
sphere and that the curves $C$ and $-C$ are equivalent on the
oriented sphere; with other words, the contribution of such a curve

to the number of orbits of oriented curves on the unoriented sphere is 2,

to the number of orbits of unoriented curves on the oriented sphere is 1,

to the number of orbits of oriented curves on the oriented sphere is 2.

The 4-tuple related to the type Sy3 is thus $(1,2,1,2)$.

Examples of curves of type Sy3 are

Because the symmetry type is the same for all curves in an orbit $K$
of Diff$(S^{2})$, one may speak of the symmetry type of $K$; we
also say that $K$ realizes a diagram when the elements of K realize
it.

If $\Gamma$ is a realizable diagram, then we associate with it the
following integers:

$v$ (resp. $w,x,y,z$) is the number of orbits of type $Sup$
(resp. $Sy1,Sy2,Sy3$, $Asy$) which realize $\Gamma$;

$uu$ (resp. $ou$) is the number of Diff$(S^{2})$-orbits of
unoriented (resp. oriented) curves which realize $\Gamma$ (the
sphere is unoriented);

$uo$ (resp. $oo$) is the number of Diff${}^{+}(S^{2})$-orbits of
unoriented (resp. oriented) curves which realize $\Gamma$ (the
sphere is oriented).

N.B.: in the symbols $uu$, …, $oo$, the first character is related
to the curves and the second one to the sphere; $u$ is the initial
of unoriented and $o$ begins oriented.

The stated properties of the five types of symmetry imply that $uu$,
$ou$, $uo$ and $oo$ are linear functions of $v$, $w$, $x$, $y$ and
$z$:

where the columns of the $4\times 5$-matrix are the
4-tuples related to the five symmetry types.

As an introduction to Sect. 10, suppose we want to determine the
numbers $uu,ou,uo,oo$ of orbits which realize the Gauss diagram
having one 5-chord and four 1-chords. The method explained in
Sect. 7 produces seven orbits under Diff$(S^{2})$; they are
represented by the curves

The first two curves have type Sup, the following
three (from left to right), types Sy1, Sy2, Sy3, and the last two, type Asy; hence $v=2,w=x=y=1,z=2$
for the studied diagram and the matrix equation yields $uu=7,ou=10,uo=10$ and $oo=16$.

Table 6 gets together the results of the enumeration of orbits of
spherical curves with 5 double points; each row corresponds to a
family, named in the first column; the second column gives the
number of Gauss diagrams of order 5 in the family; the columns 3 to
7 give for each symmetry type $T$ (in the order Sup, Sy1, Sy2,
Sy3, Asy), the number of orbits of type $T$ realizing diagrams of
order 5 in the family; the columns 8 to 11 give

the number $UU$ of orbits of unoriented curves with 5
double points on the unoriented sphere,

the number $OU$ of orbits of oriented curves with 5 double
points on the unoriented sphere,

the number $UO$ of orbits of unoriented curves with 5
double points on the oriented sphere,

the number $OO$ of orbits of oriented curves with 5 double
points on the oriented sphere.

The last row gives total numbers; for example, the number of
realizable Gauss diagrams of order 5 is 15, the number of orbits of
supersymmetric spherical curves with 5 double points is 10 (but
there is only one orbit of type Sy3), the number of
Diff${}^{+}(S^{2})$-orbits of oriented curves is 182.

Table 8: Numbers of orbits of curves with 5 double points

As expected, the last four entries of the last row coincide with the
values found by Arnold (see last column of first table in Sect. 1).

The Appendix concerning Table 6 sheds light on how to obtain the
values given in it: beside every diagram $\Gamma$ in the considered
family appear a 5-tupel giving the numbers $v,w,x,y,z$ defined
at the end of Sect. 9, and the related 4-tuple giving the numbers
$uu,ou,uo,oo$. So, the entry of this appendix describing the
realizations of the diagram considered at the end of Sect. 9 is:

Counting the orbits of spherical curves with 6 double points

The number of realizable Gauss diagrams of order 6 is 43. In the
following table, we give, for each family, the numbers of orbits of
unordered curves with 6 double points classified according to
symmetry type, and the numbers of orbits if one supposes that the
curves or the sphere are oriented.

The elements of the header of Table 9 have the same meaning as in Table 6.

Remark.

(1)

We got the last four values of the row “Total” of Table 7 in
2004 and added them to four sequences initiated by V. I. Arnold in
the On-line Encyclopedia of Integer Sequences; the identifiers are
A008989 for the
$UU$
-sequence, A008988, A008987 and AA008986 for the
other three.

(2)

The same remark, for
$n=7$
instead of
$n=6$
, is valid for
the last four values of the row “Totals” in Table 8.

(3)

In a recent paper, Robert [[Coquereaux and Zuber2016]] use another method to count the last four numbers of
the rows “Total” in Tables 8, 9 and 10 in their Table 5 (p. 25),
they confirm our results without using our 22 families.

(4)

The interested reader who wants to know about the
contributions of a specific Gauss diagram of order
6
to the
numbers
$Su,S1$
,…,
$UO,OO$
(or comparing her/his own counts with
ours) may consult the Appendix concerning Table 9.

Counting the orbits of spherical curves with 7 double points

The number of realizable Gauss diagrams of order 7 is 172 (13 are
leader-diagrams and 159 are not). Table 10 is analogous to Tables 8
and 9, giving now informations about curves with 7 double points in
each of the 22 considered family.

The contributions of a given Gauss diagram to the numbers $Su,S1$,
…, $UO,OO$ are detailed in the Appendix concerning Table 10.

Table 10: Numbers of orbits of curves with 7 double
pointsTable 11: Numbers of orbits of curves with a given symmetry type.

In Table 11, we give, for each symmetry type T, the numbers of
orbits of unordered curves of type T classified according to their
number of double points.

In a last table, we consider only curves without any simple
loop (realizations of leader-diagrams); the numbers of orbits of
such curves (oriented or not) under $\mbox{Diff}(S^{2})$ or
$\mbox{Diff}^{+}(S^{2})$) are given in Table 12. The given values also
appear in the Table 6 (p. 26) of [Coquereaux and Zuber2016].

Grateful thanks are due to F. Aicardi, J.-P. Doignon, S. Duzhin and
J.-B. Zuber who encouraged me to write and publish this work. I also
thank the referee for helpful comments on the first version of the
paper.

[Chmutov et al.2006] Chmutov, M., Hulse, T., Lum, A., Rowell P.: Plane and spherical curves: an investigation of their invariants. Proceedings of the research experiences for undergraduates program in mathematics, Oregon State University (2006)